Self-Consistent Sources and Conservation Laws for Super Tu Equation Hierarchy

Based upon the basis of Lie super algebra B(0,1), the super Tu equation hierarchy with self-consistent sources was presented. Furthermore, the infinite conservation laws of above hierarchy were given.


Introduction
Soliton equations with self-consistent sources have been receiving growing attention in recent years.Physically, the sources may result in solitary waves with a non-constant velocity and therefore lead to a variety of dynamics of physical models.For applications, these kinds of systems can be used to describe interactions between different solitary waves.Ma and Strampp systematically applied explicit symmetry constraint and binary nonlinearization of Lax pairs for generating soliton equation with sources [1].Then, Ma presented the soliton solutions of the Schrödinger equation with self-consistent sources [2].The discrete case of using variational derivatives in generating sources was discussed in [3].
With the development of soliton theory, super integrable systems associated with fermi variables have been receiving growing attention.Various methods have been developed to search for new super integrable systems, Lax pairs, soliton solutions, symmetries and conservation laws, etc. [4]- [11].In 1997, Hu proposed the supertrace identity and applied it to establish the super Hamiltonian structures of super-integrable systems [4].Then Professor Ma gave a systematic proof of super trace identity and presented the super Hamiltonian structures of super AKNS hierarchy and super Dirac hierarchy for application [5].The super Tu hierarchy and its super-Hamiltonian structure was considered [6].Recently, Yu et al. considered the binary nonlinearization of the super AKNS hierarchy under an implicit symmetry constraint [7] and the Bargmann symmetry constraint and binary nonlinearization of the super Dirac systems [8].Meanwhile, various systematic methods have been developed to obtain exact solutions of the super integrable such as the inverse transformations, the Bäcklund and Darboux transformations, the bilinear transformation of Hirota and others [9]- [11].
This paper is organized as follows.In Section 2, the method for establishing super integrable soliton hierarchy with self-consistent sources by using Lie super algebra ( ) was presented.For application, the super Tu hierarchy with self-consistent sources was obtained in Section 3. In Section 4, conservation laws of super Tu hierarchy were given.

A Kind of Super Integrable Soliton with Hierarchy Self-Consistent Sources
In the following.Consider a basis of Lie super algebra ( ) 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 , 1 0 0 , 1 0 0 , 0 0 0 , 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 e e e e e We introduce the loop algebra ( ) where the loop algebra ( ) where , From the spectral problem (3), the compatibility condition gives rise to the well-known zero curvature equation The general scheme of searching for the consistent ( ) n V and generating a hierarchy of nonlinear equations was proposed as follows [5].We solve the equation And search for , , , where ( ) We consider the super trace identity of super integrable systems [4] [5] where Str means the super trace.Defining a scalar

( )
, The sets { } m H proves the conserved densities of (4).The Hmailtonian form with 1 n H + can be written as where L is a recursion operator and J is a symplectic operator, and According to (3) and ( 5), we consider the auxiliary linear problem.For N distinct , 1, , j j N λ =  , the fol- lowing systems result from (1) Based on the results [11], we show that the following equations where j α are constants.Equation ( 11) determines a finite dimensional invariant set for the flows (9).
For (10), it is known that where Str denotes the super trace of a matrix and According to (11), for a specific 0 0 k n ≥ , we demand that ( ) ( ) From ( 9) and ( 11), a kind of super integrable hierarchy with self-consistent sources can be present as follows

The Super Tu Hierarchy with Self-Consistent Sources
The super Tu spectral problem associated with Lie super algebra ( ) where λ is a spectral parameter, q and r are even variables, α and β are odd variables [6].
Taking 0 The co-adjoint equation associated with ( 16) Then ( 17) is equivalent to 1 Which results in the recurrence relations where Upon choosing the initial conditions ≥ can be worked out by the recurrence relations (20).The first few sets are as follows: Let us associate the problem (16) with the following auxiliary problem S. X. Tao ( ) ( ) The compatible conditions of the spectral problem (16) and the auxiliary problem ( 22) are Which refer the super Tu equation hierarchy Here 24) is called the n-th Tu flow of this hierarchy.Using the super trace identity (7), we have ( ) Therefore, the super Tu soliton hierarchy Equation ( 24) can be written as the following super Hamiltonian form: where Is a super symplectic operator, and n H is given by (25).The first non-trivial nonlinear of super Tu hierarchy is given by its second flow ( ) Which possesses a Lax pair of U defined in (16) and ( ) Next we will establish the super Tu hierarchy with self-consistent sources.Consider the linear system For the system (28), we consider the in the Lie super algebra ( ) According to the results in (15), the super Tu hierarchy with self-consistent sources is presented as The first nontrivial integrable super Tu hierarchy with self-consistent sources is its second flow ( ) ( ) , it is the well known nonlinear Tu equation with self-consistent sources.So system (30) is a novel super integrable equation hierarchy.

Conservation Laws for the Super Tu Hierarchy
In what follows, we will construct conservation laws of the super coupled Burgers equation.Introduce the variables: where ( ) ( ) We expand , K G in powers of 1 λ − as follows 0 0 , where ( ) 0 j p k = , ( ) And a recursion formula for n k and n g , we derive the conservation laws of ( where 0 1 , c c are constants of integration.Then the first two conserved densities and currents are ( ) ( ) ( ) where n k and n g can be calculated from (36).The infinitely conservations laws of (36) can be easily ob- tained in (32)-(40) respectively.
is the right form of conservation laws.We expand σ and θ as series in powers of λ according with the coefficients, which are called conserved densities and currents respectively